Question

A random variable \( Y \) is normally distributed with a mean of 200 and a standard deviation of 10. Compare the following quantities: \[ \text{The probability of the event that the value of } Y \text{ is greater than 220} \quad \text{and} \quad \frac{1}{6} \]

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given.
• E.

Hint:

When dealing with normal distributions, the exact probability of an event can often be calculated using a Z-table or statistical software. Without these tools, the comparison is imprecise.

Answer 1

The Correct Option is D

Step 1: Analyze the normal distribution.
We are given that \( Y \) is normally distributed with a mean of 200 and a standard deviation of 10. The value of 220 is 2 standard deviations above the mean. In a normal distribution, the probability that \( Y \) is greater than 220 is less than 5% of the total data, but the exact value is difficult to determine without using a Z-table or normal distribution function.
Step 2: Compare the probabilities.
From the explanation, it is clear that the probability of \( Y \) being greater than 220 is less than 5%, which is less than \( \frac{1}{6} \), or approximately 16.67%. Thus, Quantity B ( \( \frac{1}{6} \) ) is greater than Quantity A.
Step 3: Conclusion.
Thus, the correct answer is that Quantity B is greater than Quantity A.
Final Answer: \[ \boxed{\text{The correct answer is (4) The relationship cannot be determined from the information given.}} \]